Evaluation of an Analytic Model for Car DynamicsM. Ruf∗†, J. R. Ziehn∗‡§, D. Willersinn∗, B. Rosenhahn‡, J. Beyerer∗† and H. Gotzig¶

∗Fraunhofer Institute of Optronics, System Technologies and Image Exploitation (IOSB), 76131 Karlsruhe, GermanyEmail: {miriam.ruf, jens.ziehn, dieter.willersinn, juergen.beyerer}@iosb.fraunhofer.de

†Vision and Fusion Laboratory (IES), Karlsruhe Institute of Technology (KIT), 76131 Karlsruhe, GermanyEmail: miriam.ruf@kit.edu

‡Institut fur Informationsverarbeitung (TNT), Leibniz Universitat Hannover, 30167 Hanover, GermanyEmail: {rosenhahn, ziehn}@tnt.uni-hannover.de

¶Valeo Schalter und Sensoren GmbH, 74321 Bietigheim-Bissingen, GermanyEmail: heinrich.gotzig@valeo.com

Abstract—This paper presents and evaluates a differentialgeometric model for single and double track vehicles, whichallows to estimate internal state parameters (steering angles,velocities, wheel speeds, ...) from a given planar trajectory overtime. Possible applications include accident reconstruction fromvideo footage, constraining trajectory optimization by physicallimits, and extracting control parameters a-priori from trajec-tories, for use in open-loop (non-feedback) controllers for shortintervals. The model is easy to compute, yet the evaluation withIPG CarMaker suggests a good level of accuracy.

I. INTRODUCTION

Vehicle models are applied in driver assistance systemsto make the behavior of the ego vehicle computable usingsimplifying assumptions to a certain degree. Commonly thesemodels are based on differential equations that allow to predictthe vehicle behavior based on the control inputs.

The model presented here for evaluation is a (single ordouble track) model with a focus on easy computation basedon differential geometry properties of the given trajectory,assuming it is twice continuously differentiable (hence thename C2 model). Applications include deriving the controlcommands from a pre-computed trajectory, or supporting thecomputation of a physically feasible trajectory (e.g. by varia-tional methods, [10]) by providing a simple way of imposinghard constraints on control inputs, such as steering angles andangular velocities, or vehicle speeds and accelerations. Theapplicability of the model for both purposes will be evaluatedin this paper.

II. STATE OF THE ART

Depending on the desired level of detail and the systeminformation available, there exist different approaches for mod-eling vehicle dynamics [5], which can be used for predictingthe behavior—and therefore the future trajectory—of an egovehicle. The most common method to plan this trajectory is touse differential equations to model the influence of the controlparameters on the vehicle state ([7], [1], [4], [6]), and therebypredict the trajectory given a set of control parameters. Thisapproach is advantageous for assuring that only valid controlparameters are used in optimization, and—depending on thevehicle model—it can be very accurate. Yet, the computationaleffort is high and modeling can be difficult because trajectory

§Corresponding author

planning is performed by manipulating the inputs, not thetrajectory itself. To limit the computational effort, simplifiedmodels of vehicle dynamics are used.

To the knowledge of the authors, thus far no detailedmodel for analyzing a positional trajectory for state and controlparameters has been presented and evaluated for automotiveapplications.

III. MODEL DERIVATION

In this section the presented model will be motivated andderived geometrically.

A. Vehicle Model

The vehicle model in this paper can be a single trackor double track model. The following vehicle parameters areutilized in the model:

• The wheel base l, which is the distance between thefront and the rear axle.

• The half width h of the track, which is half the distancebetween wheels on the same axle.

• The tire radii rfront and rrear at the respective axes.

The vehicle state is taken to be comprised entirely of

• its position ξ = [ξx, ξy]T, which is taken to lie on thecenter of the rear axle (cf. Fig. 1),

• its velocity dξ/dt = ξ = [ξx, ξy]T, which alwayspoints into the heading direction ψ,

• its steering wheel angle δw, which maps to the leftand right tire angles δleft, δright, and to the mean tireangle δ = (δleft + δright)/2, and

• its tire rotation angles around the car axes, for thefront left, front right, rear left and rear right tiresρfl, ρfr, ρrl, ρrr.

Notable simplifications, some of which are incorporated inthe single track model [8], include:

• The tire slip is assumed to be zero at all tires at alltimes, both for lateral and longitudinal slip.

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2014 International Conference on Mechatronics and Control (ICMC) July 3 - 5, 2014, Jinzhou, China

978-1-4799-2537-7/14/$31.00 ©2014 IEEE

• Bank (roll) and pitch angles of the vehicle are notconsidered.

• The vertical axis is not considered; the center ofgravity lies in the 2D plane.

• Wheel loads and varying radii are not considered.

• Any internal structure, such as engine and gears, isomitted. (For vehicle control, a drive-by-wire systemis assumed to be in place that operates based ondesired steering wheel angles δw and accelerations a.)

B. Trajectories

A trajectory is taken to be the planar curve described by acar’s reference point, which lies on the center of the rear axle(Fig. 1). This planar curve is defined by

ξ : T → X × Y (1)

where T is an interval of time, and X and Y representCartesian coordinates on the surface on which the ego vehiclecan navigate. Further, ξ must lie in C2 (i.e. be twice continu-ously differentiable), to assure that all parameters establishedhere are well defined. If the surface has a non-Cartesianparametrization (for example due to curvature), the modelderived in this section is invalid and Riemannian geometrymust be applied [9, p. 127ff.], which is beyond the scope ofthis paper. However, it is assumed that taking the trajectoryto lie on a Euclidean plane represents a suitable simplificationfor a wide range of practical applications.

C. State and Control Parameters

Derivatives of the trajectory will be denoted ξ = dξ/dtand ξ = d2ξ/(dt)2. For any point along the trajectory, thereis a base of R2 consisting of a tangent vector and a normalvector [3, p. 19]

T =ξ

‖ξ‖and N =

[0 −1

1 0

]T . (2)

The tangent vector must be continuous for the curvature(required later, in (5)) to be finite. Where T cannot be directlycomputed via (2), because ξ(t) ≡ 0 on any interval I , its valueon I can be set to either of the outer limits for T on I . Theselimits should be identical, otherwise again the curvature willbe infinite. If such limits exist for no choice of I , then the caris stationary (ξ(t) ≡ 0 for all t) and no directional parameterscan be derived.

If the car’s coordinate system is identified with the (T ,N)base, the longitudinal speed and acceleration can thus beobtained as the projection of total velocities and accelerationsonto the tangential vector:

v =⟨T∣∣∣ ξ⟩ =

⟨ξ∣∣∣ ξ⟩‖ξ‖

= ‖ξ‖ and (3)

a =⟨T∣∣∣ ξ⟩ =

⟨ξ∣∣∣ ξ⟩‖ξ‖

, (4)

where the identification of bases makes use of the assumptionthat the slip be zero (i.e. 〈N |ξ〉 ≡ 0 at all times). It should

δleft

ξ(t)

NT

Rrear

h

Rleftrear Rleft

front

l

Fig. 1: Ackermann steering geometry in a constant curvature turn. Theindividual trajectories of every point on the car are concentric circles or varyingradii. All tires are aligned tangential to their respective circles. The steeringangles δw and δs effect the turn radius Rrear of the car’s reference point alongξ. In a right turn, Rrear < 0, so Rleft

rear = Rrear − h produces a larger absoluteradius.

also be noted that (4) is not ‖ξ‖ unless the vehicle is movingstraight ahead.

The curvature is defined (for arc length s; [2], [3]) by

κ =⟨

ddsT

∣∣N⟩ =det[T ,T ]

‖ξ‖=

det[T , ξ]

‖ξ‖2=

det[ξ, ξ]

‖ξ‖3(5)

and specifies the circular approximation of the trajectory atevery point: The circle that approximates ξ lies in the directionof N off the trajectory at a distance and radius of κ−1 =Rrear, the turn radius of the ego vehicle’s reference point (cf.Fig. 1) (not to be confused with the rear tire radius, rrear). Bothcurvature and radius are negative for right turns (see Fig. 1).

The turn radius of the car is effected by the front tire angles,δs, with s ∈ {left, right}, and (indirectly) the steering wheelangle δw. Figure 1 shows the geometry that is used to derivethe front left tire angle δleft from the rear axle turn radius Rrear(described by the trajectory ξ). Since all four tires describeconcentric turn circles (albeit at four different radii Rs

rear andRs

front), and since their direction of motion is tangential to theirrespective circles, the right triangle in Fig. 1 can be established.The effect of the front tire angle δs on the rear tire turn radiusRs

rear is mediated by the wheel base l and the half track widthh via:

δs = arctan

(l

Rsrear

)= arctan (l κs)∣∣∣ ∣∣∣

= arctan

(l

Rrear ±s h

)= arctan

(l κ

Rrear

Rsrear

) (6)

(where κs is the curvature of the individual front tire trajecto-ries, and ±left = −,±right = +) using the curvature calculatedin (5). It should be noted that the use of atan2 is not necessaryhere since the tire angles are certainly limited to ±π/2.

Using the tire radii rfront and rrear, the angular velocities ofthe tires can be obtained as

ρrs =‖ξ‖rrear· R

srear

Rrear=‖ξ‖rrear· Rrear ±s h

Rrear(7)

and

ρfs =‖ξ‖rfront

· Rsfront

Rrear=‖ξ‖rfront

·√l2 + (Rrear ±s h)2

Rrear. (8)

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reference maneuver

sim(Fig. 3)

ξ1 a1, δ1, ...

aC2 , δC2 , ...

sim(Fig. 4)

ξ2 a2, δ2, ...

C2

IV-A

IV-B1IV-B

Fig. 2: Structure of the evaluation of the C2 model: A reference maneuveris passed to CarMaker for simulation. The simulation yields a trajectory ξ1and internal parameters a1, δ1, etc. The trajectory is passed to the C2 model.The resulting estimated parameters are compared directly in Figs. 9 and 10.Accelerations and steering wheel angles are further used as control inputs toanother simulation, which again yields a trajectory ξ2 and internal parametersa2, δ2, etc. The obtained results are discussed in the sections indicated bydashed arrows.

IPG Driver CarMakerDynamic

Simulation

refer-encema-

neuver

δw

brake

gas

ξ1

a1

δ1

. . .

Fig. 3: Structure of the first simulation from Fig. 2. A given referencemaneuver is passed to the CarMaker “Driver” subsystem, which finds controlparameters δw, brake and gas to execute the reference maneuver as closely aspossible. These inputs are then passed to the CarMaker Dynamic Simulationof the actual car, yielding the outputs described in Fig. 2.

The heading direction ψ, or yaw, can be specified relativeto a “north” direction n as

ψ = ](T ,n) = ](ξ,n) = atan2(

det[n, ξ], 〈n|ξ〉), (9)

and its derivative, the yaw rate, is thus

ψ =det[ξ, ξ]

‖ξ‖. (10)

IV. MODEL EVALUATION

The properties defined above can be computed for everypoint along any twice continuously differentiable (non-static)trajectory. This allows to transform a surface curve into controland state parameters for a simple vehicle model. In this section,the model accuracy for two possible applications will behighlighted.

A. True Maneuver vs. C2 model Analysis

This is arguably the most conclusive comparison. Anobserved trajectory ξ1 is analyzed by the C2 model to estimatethe state variables and control parameters that were attainedduring the maneuver. Common purposes of this step would bethe following:

C2 model CarMakerDynamic

SimulationSimulink

model

δ 7→ δw

×

i

ξ1

δC2 δwC2

aC2

brk.

gas

ξ2

a2

δ2

. . .

Fig. 4: Structure of the second simulation from Fig. 2. A given trajectory ξ1 ispassed to the C2 model to obtain parameters aC2 , δC2 . Desired accelerationsaC2 are passed to an IPG ACC model included in CarMaker, which findscontrol parameters brake and gas to achieve these accelerations. The meantire angle δC2 is turned into steering wheel angles δw

C2 by a linear factor i,as a simplification. These inputs are then passed to the CarMaker DynamicSimulation of the actual car, yielding the outputs described in Fig. 2.

• “Reverse-engineering” vehicle states from e.g. a videocamera or on-board recording of positions.

• Constraining trajectory optimization processes tophysically possible states.

In this evaluation, a BMW 118i is modeled in the simulationsoftware IPG CarMaker. The simulation is performed at a rateof 1 000 Hz in CarMaker, the output values as passed on tothe C2 model and used in the evaluation are sampled at a rateof 100 Hz.

1) Sine Steering: To evaluate the robustness to steering,the simulated BMW drives at a constant goal speed of v ∈{6 km/h, 30 km/h, 50 km/h} and steers with a steering wheelangle of δw = A · sin(2πt/τ), A ∈ [0, π], t ∈ [0 s, 40 s], τ ∈[1 s, 20 s]. The true state variables from CarMaker, a1, δw

1 , ...are compared to the estimates from the C2 model, aC2 , δw

C2 , ...using the error metric (for any state quantity x)

Ex =

√1

40 s

∫ 40 s

t=0 sdt(x1(t)− xC2(t)

)2. (11)

The error in Fig. 5b shows, that the mean tire angleaccuracy deteriorates with increasing frequency and amplitudeA of the steering wheel angle, as would be expected. The orderof errors is approximately 0.2 rad, with the error increasing ap-proximately linearly with frequency, and slightly superlinearlywith the steering wheel angle.

Contrary to this, the errors in velocity (Fig. 5a) and accel-eration show a salient pattern varying over the period length.This pattern correlates with the car’s bank angle error andcan be attributed to a violation of the models assumptions forthe vehicle to remain level at all times. Remarkably the errordecreases for some period lengths at increasing amplitudes,suggesting that parametrization errors, possibly in the tire radii,are counteracted by the errors due to bank angles.

2) Race Track: To evaluate the model in a realistic combi-nation of various maneuvers, the simulation is applied to theHockenheimring race track in Germany (a map can be foundin Fig. 7, black line). The race track has a length of 4.574km and is driven in the clockwise sense. In the first step, the

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0.513

0.532

1 s 20 s

(a) Ev in [m/s]

0.001

0.246

0 A

(b) Eδ in [rad]

0.001

0.246

1 Hz 20 Hzf

(c) Eδ in [rad]

Fig. 6: Errors as shown in Fig. 5, as families of curves. (c) is plotted over f instead of τ , to reveal the linearity.

0 A π

0 s

τ

20 s

0.532

0.513

(a) Ev in [m/s]0 A π

1 s

τ

20 s

0.246

0.001

(b) Eδ in [rad]

Fig. 5: Errors obtained at the sine steering example (Sec. IV-A1) at v = 50km/h.

CarMaker module IPG Driver is used to automatically drivealong the track at maximum speed; a top speed of 170 km/his attained. The comparison of true parameters with the C2

model estimates over the path length can be found in Fig. 10,the correlation between them is shown in Fig. 9. The latterstates the mean error µ and the standard deviation σ of theerror. It furthermore states the slope of the linear least squaresapproximation of the measurements, namely the pseudoinversesolution m to

x1(t) ·m = xC2(t)

for any state quantity x over all measurement times t, whichyields the m to minimize

∫dt (x1(t)·m−xC2(t))2, consistently

with (11). This m indicates a systematic factor between theestimated value xC2 and the ground truth x1.

The velocities (Fig. 9b) show an excellent correlation, theaccelerations (Fig. 9a) deviate more notably, partly due to dis-cretization. Figs. 9e and 9f show mostly a good correlation, butalso the effects of the violated assumption of zero longitudinalslip: The small spikes to the left and right occur, when eitherthe brake, or the gas pedal is pushed notably. The tire speedchanges almost instantaneously (the BMW has a rear-wheeldrive), while due to slip, the vehicle speed does not reactwithout delay. During this delay the spikes of high deviationform, since the C2 model cannot derive the change in wheelspeed from the vehicle speed. The front wheels, which are notconnected to the drivetrain, are not subject to these effects.

The tire steering angles (Fig. 9c–f) exhibit a slightly curvedtrend, which can stem from errors in wheel base l and halftrack width h.

Figure 10d shows an extract of the tire angle where δC2

deviates notably from δ1. The extract corresponds to the slightand elongated left bend between 1 km and 2 km (cf. Fig. 7).The C2 model estimates a steering angle that would follow thebend at perfect grip. The simulated BMW however passes thebend in a drift/slip state, with a notable lateral speed 〈ξ|N〉 ≈−3 km/h, as shown in Fig. 10e (however at longitudinal speed

4 km

3 km

2 km1 km

n

0 km4.574 km ξ1

ξ2

Fig. 7: Hockenheimring (true shape ξ1, black) traversed with an open-loop controller (ξ2, white) based only on precomputed steering angles andaccelerations extracted from a normal traversal ξ1 via the C2 model. Themilestones (in kilometers) are shown as dots.

〈ξ|T 〉 ≈ 170 km/h), and is thus oversteering. Since lateral slip,and thus lateral speed is not considered in the C2 model, thetire steering angle δC2 is lower (due to a lack of oversteering).

B. C2 Model Input vs. Execution of Results as Open-LoopControl Parameters

In this comparison, it is shown how a given trajectory ξ1can be analyzed by the C2 model to find control parametersaC2 , δw

C2 for an open-loop (non-feedback) controller. The re-sulting trajectory ξ2 is compared to the original ξ1.

The analysis in this step is less conclusive than the onein the previous Section IV-A, since a major factor is howwell the ego vehicle (in this case the IPG ACC along withthe assumption of a constant steering ratio i ≈ 18.4) ex-ecutes the control parameters aC2 and δw

C2 . Therefore, thisevaluation essentially demonstrates a lower bound of quality,which can be achieved by a very simple control model. Moresophisticated models would likely be able to reproduce thedesired parameters aC2 , δC2 much more closely (while still notusing feedback). Figure 7 shows the positional deviation thatwould occur if the entire Hockenheimring is navigated via theoffline control parameters aC2 , δw

C2 by an open-loop controller.Naturally, navigating approximately 5 km without feedbackbased on predetermined parameters is bound to produce asignificant error, however the resemblance between ξ1 and ξ2is notable. An evaluation that divided the Hockenheimring intoincreasingly small segments and measured the mean end pointdeviation per segment length converged towards an order of3 cm/1 m.

1) C2 Model Output vs. Execution of Results as Open-Loop Control Parameters: Even though the comparison ofthe C2 model outputs and the original control parameters of

2449

ξ1 show promising results, the second simulation shows asignificant offset, as discussed above. To convey an impressionof how the deviations in the steering ratio and the drive-by-wire ACC used in the second simulation affect the quality ofthe results (compared to influences due to the simplificationsin the C2 model), Fig. 8 compares the velocities vC2 and v2on a representative extract along the time axis. What can beseen is a delay of approximately 0.7 s between desired inputacceleration and the ACC reaction, however for decelerationsthe delay is only 0.1 s. This asymmetry results in a peak dif-ference of about 3% between desired and achieved velocities,and also to a misalignment between delayed velocities, nearlyundelayed steering angles (delay below 0.01 s) and the actualtrack position. Since both steering wheel angle magnitudes andreaction times match the desired values closely, they are notdepicted. This implies that the main source of error betweenξ1 and ξ2 are the simple ACC and steering ratio, which arelikely outperformed by state-of-the-art drive-by-wire systems.

V. CONCLUSION

In this paper, the C2 model was derived geometricallyand applied to different maneuvers for different purposes inthe simulation tool CarMaker by IPG. It could be shown thatthe analytic possibilities of this model are adequate. A giventrajectory can be analyzed for the internal state parameters ofa vehicle with an accuracy of more than 99.9% for velocitiesand less than 99.7% for steering angles. This may allowfor applying the C2 model for purposes like reconstructingaccidents or constraining trajectory optimizations.

A second purpose of the C2 model was to use the stateparameters a, δ (acceleration and mean tire steering angle) asinputs to a drive-by-wire system. Due to the simplicity ofthe drive-by-wire model, the results merely present a lowerbound of quality. Still a deviation of just 3 cm/1 m per drivenlength was be achieved for a simple open-loop (non-feedback)controller.

The results are promising; however it is necessary toproduce a more sophisticated evaluation of the C2 model.A wider range of maneuvers must be evaluated to assess theanalytic potential of the C2 model; also, real measurements ofactual vehicles must be used to complement the simulated datato study the influence of uncertainties. Furthermore, tests withmore realistic drive-by-wire systems are necessary to deter-mine the degree of confidence, by which C2 model outputs canbe used as open-loop control inputs to reproduce a trajectory.These tests need to comprise both simulations with moresophisticated models, and tests in actual vehicles equipped withdrive-by-wire functionality. A significant increase in quality ofthe C2 model predictions is anticipated, however a quantitativeestimate is not possible at this stage.

ACKNOWLEDGMENT

This work was partially supported by Valeo Schalterund Sensoren GmbH within the V50 project, and by theFraunhofer-Gesellschaft along with the state of Baden-Wurt-temberg within their joint innovation cluster REM 2030.

0 5 10 15 20 25 30 35 40 4510

20

30

Fig. 8: vC2 (white) and v2 (black) in [m/s] over t in [s], showing the effectof the delay of the drive-by-wire ACC.

REFERENCES

[1] E. Bauer, F. Lotz, M. Pfromm, M. Schreier, B. Abendroth, S. Cieler,A. Eckert, A. Hohm, S. Luke, P. Rieth, V. Willert, and J. Adamy.PRORETA 3: An Integrated Approach to Collision Avoidance andVehicle Automation. at – Automatisierungstechnik, 60(12):755–765,2012.

[2] B. Dubrovin, A. Fomenko, S. Novikov, S. Novikov, and R. Burns.Modern Geometry - Methods and Applications: Part 3: Introduction toHomology Theory. Graduate Texts in Mathematics. U.S. GovernmentPrinting Office, 1990.

[3] J. Eschenburg and J. Jost. Differentialgeometrie und Minimalflachen.Springer London, Limited, 2007.

[4] D. Ferguson, T. M. Howard, and M. Likhachev. Motion Planning inUrban Environments. In M. Buehler, K. Iagnemma, and S. Singh, edi-tors, The DARPA Urban Challenge, volume 56, pages 61–90. SpringerBerlin Heidelberg, 2009.

[5] R. Isermann. Fahrdynamik-Regelung: Modellbildung, Fahrerassisten-zsysteme, Mechatronik. Friedr. Vieweg & Sohn Verlag, Wiesbaden,2006.

[6] M. Montemerlo, J. Becker, S. Bhat, H. Dahlkamp, D. Dolgov, S. Et-tinger, D. Haehnel, T. Hilden, G. Hoffmann, B. Huhnke, D. Johnston,S. Klumpp, D. Langer, A. Levandowski, J. Levinson, J. Marcil, D. Oren-stein, J. Paefgen, I. Penny, A. Petrovskaya, M. Pflueger, G. Stanek,D. Stavens, A. Vogt, and S. Thrun. Junior: The Stanford Entry in theUrban Challenge. In M. Buehler, K. Iagnemma, and S. Singh, editors,The DARPA Urban Challenge, volume 56, pages 91–124. SpringerBerlin Heidelberg, Berlin and Heidelberg, 2009.

[7] S. Schmidt and R. Kasper. Ein hierarchischer Ansatz zur optimalenBahnplanung und Bahnregelung fur ein autonomes Fahrzeug. at –Automatisierungstechnik, 60(12):743–754, 2012.

[8] D. Schramm, M. Hiller, and R. Bardini. Modellbildung und Simulationder Dynamik von Kraftfahrzeugen. Springer Berlin Heidelberg, Berlinand Heidelberg, 2010.

[9] D. J. Struik. Lectures on Classical Differential Geometry. DoverPublications, 2nd edition, 1961.

[10] B. Van Brunt. The Calculus of Variations. Springer, 2010.

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−8

−4

±0

+4

+8

−8 −4 ±0 +4 +8

µ = +0.0174

σ = 0.1287

m = 0.9588

(a) a1 7→ aC2 , [m/s2]

10

20

30

40

50

10 20 30 40 50

µ = +0.1880

σ = 0.0003

m = 1.0000

(b) v1 7→ vC2 , [m/s]

0 100 160

100

160 µ = −0.2092

σ = 0.3751

m = 1.003

(c) ρfl1 7→ ρfl

C2 , [rad/s]

0 100 160

100

160 µ = −0.2127

σ = 0.3681

m = 1.003

(d) ρfr1 7→ ρfr

C2 , [rad/s]

0 100 160

100

160 µ = +0.3189

σ = 0.6976

m = 0.9950

(e) ρrl1 7→ ρrl

C2 , [rad/s]

0 100 160

100

160 µ = +0.2987

σ = 0.6544

m = 0.9955

(f) ρrr1 7→ ρrr

C2 , [rad/s]

−0.2 0 +0.2

−0.2

0

+0.2µ = −0.0022

σ = 0.0028

m = 0.9276

(g) δleft1 7→ δleft

C2 , [rad]

−0.2 0 +0.2

−0.2

0

+0.2µ = +0.0015

σ = 0.0026

m = 0.9594

(h) δright1 7→ δright

C2 , [rad]

Fig. 9: Correlation plots of the vehicle parameters along the Hockenheimringtrack. The vertical axis always refers to the C2 model estimates, the horizontalaxis shows the true data. Mean error µ and standard deviation σ are giveninside the plots along with the slope m of the least squares linear approx-imation, to indicate systematic scale errors. (a): Acceleration. (b): Velocity.(c)–(f): Rotation speeds for each of the wheels. The arrows in (e) and (f)indicate errors due to longitudinal slip when either brake (peak to the left) orgas pedal (peak to the right) are pushed notably. (g), (h): Steering angles forthe front wheels.

0 km 1 km 2 km 3 km 4 km

10

20

30

40

50

(a) v = 〈ξ|T 〉 in [m/s]

0 km 1 km 2 km 3 km 4 km

−5

0

+5

(b) a in [m/s2]

0 km 1 km 2 km 3 km 4 km

−0.2

−0.1

0.0

+0.1

(c) δ in [rad]

0.00

0.01

0.02

1 km 1.5 km 2 km(d) δ in [rad]

−1.0

−0.5

0.0

1 km 1.5 km 2 km

(e) 〈ξ|N〉 in [m/s]

Fig. 10: Parameters extracted along the Hockenheimring (white) compared totrue simulation data (black). (d) and (e) show the effect of oversteering that isnot modeled, leading to underestimated steering angles during slip. The C2

model estimate in (e) is zero, since slip is not modeled.

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